Iriyama, Ohya, and Volovich have developed a model of a linear quantum Turing machine (LQTM). This is a generalization of a classical QTM that has mixed states and that allows irreversible transition functions. These allow the representation of quantum measurements without classical outcomes.[4]

A quantum Turing machine with postselection was defined by Scott Aaronson, who showed that the class of polynomial time on such a machine (PostBQP) is equal to the classical complexity class PP.[5]

The input and output symbols Σ{\displaystyle \Sigma } are usually taken as a discrete set, as in the classical system; thus, neither the input nor output to a quantum machine need be a quantum system itself.

The set F{\displaystyle F} of final or accepting states is a subspace of the Hilbert space Q{\displaystyle Q}.

The above is merely a sketch of a quantum Turing machine, rather than its formal definition, as it leaves vague several important details: for example, how often a measurement is performed; see for example, the difference between a measure-once and a measure-many QFA. This question of measurement affects the way in which writes to the output tape are defined.